Rate of Temperature Increase 2

[Haf]

Suppose you have a material that you are heating with a radiative heat flux. You know the power density of the heat flux and the area it operates over. You want to calculate how long it takes for a given volume of that material to heat to a given temperature. Given that time (and the known parameters of the heat flux) you can also calculate the energy required to reach that temperature.

If the power density (W/m2) of the heat flux is too low, you reach steady state temperature before the desired temperature. As you increase power density, you reach the desired temperature before steady state temperature at some given time. At still higher power densities, you reach that temperature faster. Also, less energy is required, because less heat is conducted away from the volume you are interested in. Finally, at some relatively high power density, the effect from heat conduction away from the given volume is neglible, because the volume heats so quickly that there is literally not enough time for conduction.

Finally, here are my questions. It seems to me that once heat conduction becomes negligible, it should take the same amount of energy to heat that volume of material to the desired temperature. Does that make sense? If that's true, further increasing the power density of the heat flux should reduce the time required to reach the desired temperature in a proportional fashion (since energy is simply power multiplied by time). Is that the case? Is there some property (maybe heat capacity) that would enter in as a limitation as to how fast a volume of material can heat up?

I guess at some level, heat conduction does enter in, depending on how deeply into the material the heat flux penetrates. Does anyone have thoughts on this?

Also, at what time scales does heat conduction typically become negligible? Microseconds? Nanoseconds?

Any help would be appeciated!

Haf

 

[IRstuff]

Your use of english "negligible" is at odds with your question. If thermal conductivity is low or zero, it would take forever to heat up, except at the immediate surface, will get hot instantaneously.

If thermal conductivity is high, then the object will heat up within some reasonable time.

But your timescale is not in this universe. Most material and large objects require at least seconds or tenths of seconds to reach large temperature changes. Consider how long it takes to freeze things in your freezer.

TTFN

 

[poetix99]

I suspect, from his other posts, that IRstuff could tell you some useful information about radiative heat transfer - IF you had a well-framed question.

You have made certain assumptions that are either: categorically incorrect, or in some cases probably incorrect, depending on unspecified details of your problem.

To be a little more specific:
1) Radiative heat transfer depends upon radiant temperature(s), absorptivities, emissivities of all surfaces. Some of these properties are temperature dependent.

2) You seem to be describing a fairly complex, transient problem. Good luck.

3) Radiative h.t. is a surface phenomenon. It is the very same conduction that you wish to neglect that will actually allow any VOLUME of material to be heated.

4) Conductive h.t., especially when it can be approximated as one dimensional (as might initially be the case in your unspecified geometry), is simple to calculate. You might be able to get an order-of-magnitude grasp of what is governing or limiting in your problem. I am guessing that you want to compare conduction that is normal to the absorbing surface(s) with conduction that is lateral to those surfaces (or something like that).

5) What about convective losses?

 

[Haf]

IRstuff and poetix99,

Thanks for your responses. I appreciate your input and apologize for framing my question so poorly. Part of it was to avoid writing a three page question, though. So here goes...

First, let me be clear that the timescales and length scales I am dealing with are microseconds and microns, respectively. That complicates things a bit.

The heat flux is supplied by a laser with a 35 micron spot size. I am heating the given volume of the material from room temperature to about 300 deg. C. As you can probably guess, my power density is extremely high (it heats the material to 300 deg. C in less than 10 microseconds).

I agree that in simple cases, radiative heat transfer is a surface phenomenon. In reality, radiative heat transfer can be characterized by an extinction coefficient that can be used to calculate how deeply the heat flux "penetrates" a given material. For nearly opaque surfaces, the penetration depth is relatively small. I'm not exactly sure what the penetration depth is in my case (I can't devise a practical method for experimentally measuring penetration depth in my application), but I suspect it is near 30-50 microns. Also, I have a rough idea of what the absorptivity is.

The laser light is delivered to the material through a window, so there are no convective losses. There is, however, conduction both into the material and back through the window (as well as radially in the material and window).

So let me be clear and separate my question into two distinct parts.

First, consider the volume that is heated directly by the laser light (the volume determined by the 35 micron spot size and 30-50 micron penetration depth). Is there a limit to how fast this volume can heat up? IRstuff, you contend that the surface heats instantaneously, but that cannot be the case. It takes a finite amount of time, albeit a small amount of time. Another way to ask my question is this: suppose you keep raising your power density higher and higher. Does it make sense that at some power density, the rate of heat increase in the volume cannot be any faster, even if the power density is increased further? If so, what is the property (heat capacity?) of the material that determines how fast it can be heated in this manner (I assume it's an intrinsic material property)?

Now for the second part of the question: in 4-8 microseconds, is there enough time for heat to be conducted away from the volume of material heated directly by the radiative heat flux? I realize this depends on thermal conductivity, but in 4-8 microseconds, it doesn't seem like the heat can conduct an appreciable distance, even if the material was copper.

Now let me explain why I am asking these questions. I have conducted numerous experiments with different materials, different spot sizes, and different power densities. I seem to be running into a limit as to how fast the material will heat up to the desired temperature. I am trying to determine what that limit is controlled by. One theory I have is I am running into a limit controlled by how fast the material can heat directly from the radiative flux. The other theory I have is that the limit arrises from conduction into the material to heat some volume bigger than the volume heated directly by the laser. Unfortunately, I don't know exactly what volume of the material I need to heat to the desired temperature. I have tried to devise a way of determining that volume (I know that it is relatively small), but I can't come up with a practical method.

What I am hoping is that someone here can at least give me order of magnitude estimates as to the relevant timescales. For example, if radiation can heat the volume (35 micron diameter by 30-50 micron length) in nanoseconds or even picoseconds, then I know the limit is from conduction (this could also help me determine the volume I need to heat by using theoretical methods instead of experimental).

Sorry for the long explanation, but I hope this clears things up a bit. Thanks in advance for your input.

Haf

 

[IRstuff]

I'm not sure what types of materials you are using, but the absorption depth for for most metals are measured in the 100's of angstroms, and is nowhere near 30 microns. The absoprtion depth of 3-5 micron IR in InSb is barely 5 microns. Semi-transparent metal coatings are almost all less than 1 micron thick due to absorption. My suspicion is that you are actually limited sole by thermal conduction below a micron or so and that you are limited at the upper end by ablation of the material at the surface.

I agree convection at the time scales you are talking about is negligible, based on the thermal conductivity and specific heat necessary for convection to do much.

Conduction is limited by both specific heat and thermal conductivity. If you crank through a Fick's law calculation of transient heating of your substrate, you might be able come up with an answer. One unknown is the temperature dependence of specific heat and thermal conductivity. This could be a big impact

TTFN

 

[JKEngineer]

I am going to shoot from the hip. (Saying it first is supposed to make it ok, right? ;-) )(And it's late at night.)

First, to address the time/size/conduction importance. If we look at a large object with a large spot, then isn't the importance of the conduction in a comparably larger time scale the same as in the small spot, small object and small time scale? My point is that the trasnfer phenomena are what they are, not what the size&time combination make them if both size and time are small or large. If size were small and time large, or size large and time small, I think I would feel more comfortable dismissing some of the phenomena as insignificant.

Second, the amount of energy (=power x time) needed to raise the temperature of a given volume, if treated as net absorbed and retained energy, is a function of the material properties, not the heating method. Without phase change it's specific-heat x mass or specific-heat x volume x density. The difference between gross absorbed heat and the heat that leaves the volume is the net absorbed. The heat that leaves the volume is a combination of radiated back out, conducted away, and convected away. If you wish to argue that the molecular motion of convection does not respond fast enough to be a factor in the heat removal at the time spans of interest, then I have no basis for arguing otherwise. You may wish to do the same for conduction, but I would be less accepting, although still without data. If convection is indeed dismissed, you would still need to treat conduction through surrounding fluids, since even without gross motion, they will still carry heat as though they were solids.

I feel more comfortable talking about an average temperature in your volume rather than a single, uniform temperature, as you seem to be. Even if your penetration and absorption phenomena are completely uniform over the pentration depth, the conduction from the edges of the volume, and radiation from the free surface of the volume should not be the same, at a given time, as conduction interior to the volume, for example.

I tend to keep things simple and may have done so a little too much here. But it takes me to thinking that, even with a finite penetration depth, and one in which the power absorption is uniform with depth, there are non-uniform phenomena that will start occuring immediately (what ever that is on your time scale) and which will cause non-uniform heating of the irradiated volume.

I'll stop now, the inside of my hat is getting too wet for me to wear it.

Does this make any sense?

Jack Jack M. Kleinfeld, P.E. Kleinfeld Technical Services, Inc.
Infrared Thermography, Finite Element Analysis, Process Engineering

http://www.KleinfeldTechnical.com